Catalyst Deactivation by Active Site Coverage and Pore Blockage

Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979

D = diffusion coefficient d, = thickness of the rth section of composite membrane F ( w ) = the ratio of the maximum amplitude of the output response r(t)to the maximum amplitude of the input signal G ( t ) , eq 13 G(t) = concentration change studied, input signal H(w) = the angular difference between the sinusoidal input G ( t ) and the output I'(t), eq 14 K = exponent in exponential function, eq 16 k , = constant of the rth section of composition membrane k - oxygen mass transfer coefficient kS.=proportionality constant between oxygen flux and ampere reading L, = characteristics of liquid film resistance defined in text M = ampere reading of oxygen probe Mo = limiting value of M for t m M = steady state probe reading in gas = steady state probe reading in liquid qnr = abbreviation defined in Table I r = radial coordinate ro = radial coordinate at cathode surface s = number of regions of the composite membrane t = real time u = metavariable = u for planar cathode u = metavariable = ru for spherical cathode V = volume of vessel = volumetric gas flow rate u = oxygen concentration inside the membrane x = linear coordinate ynr = abbreviation defined in Table I z = metacoordinate = x for planar cathode z = metacoordinate = r for spherical cathode z, = metacoordinate at surface of cathode 6 = metacoordinate at surface of membrane CSAV = Czechoslovak Academy of Sciences, Czechoslovakia

-

4

245

WTW = Wissenschaftlich-Technische Werkstatten G.M.B.H., Weilheim, F.R.G. YSI = Yellow Springs Instrument, Texas Greek Letters p,, = positive roots of equation: p cot p + L, = 0 (P = oxygen solubility constant in membrane w = frequency of sinusoidal input p o = zeroth moment defined in text r = normalized probe response to a concentration change G(t) r - experimentally determined course of =-normalized probe response to a step concentration change r E 1 = experimentally determined course of Literature Cited Bandyopadhyay, B., Humprey, A., Biofechnol. Bioeng., 9, 533 (1967). Benedek, A. A., Heideger, W. J., Wafer Res., 4, 627 (1970). Carsbw, H. S.,Jeager, J. C., "Conduction of Heat in Solids", 2nd ed,Chrendon Press, Oxford, 1960. Heineken, F. G., Biotechnol. Bioeng., 12, 145 (1970). Kok, R., Zajic. J. E., Biotechnol. Bioeng., 17, 527 (1975). Lee, Y. H., Tsao, G. T., Wankat, P. C., Ind. fng. Chem. fundam., 17, 59 (1978). Linek, V., Bene& P., Biotechnol. Bioeng., 19, 741 (1977). Linek, V., Bene:, P., Biotechnol. Bioeng., 20, 903 (1978). Linek, V., Benes, P., Hovorka, F.,Vacek, V., Collect. Czech. Chem. Commun., 43, 2980 (1978a). Linek. V., Bene:, P., Sinkule, J., Kiivskv, Z., Ind. fng. Chem. Fundam., 17, 298 (1978b). Linek, V., Vacek, V., Biotechnol. Bioeng., 18, 1537 (1976). Linek, V., Vacek, V., Biotechnol. Bioeng., 19, 983 (1977). Linek, V., Vacek, V., Biotechnol. Bioeng., 20, 305 (1978). Linek, V., Stejskal, J., Sinkule, J., Vacek, V., Collect. Czech. Chem. Commun., in press, 1979. Lundsgaard, J. S.,Gronlund, J., Degn, H., Biotechnol. Bioeng.. 20, 809 (1978). Merta, K., Dunn, I. J., Biotechnol. Bioeng., 18, 591 (1976). Lightfoot, E. N., Appl. Microbial., 15, 674 (1967). Mueller, J. A., Boyle, W. C., PBca. J., Ettler, P., GrBgr, V., J . Appl. Chem. Biotechnol., 26, 309 (1976).

Received for revieu: May 11, 1978 Accepted February 13, 1979

Catalyst Deactivation by Active Site Coverage and Pore Blockage Jean W. Beeckman and Gilbert F. Froment" Laboratorium voor Petrochemische Techniek, Rijksuniversiteit, Gent, Belgium

The deactivation of porous catalysts by side reactions leading to "coke" is explained in terms of active site coverage and pore blockage. The theory is developed under the assumptions that there are no diffusional limitations inside the catalyst and that the growth of a coke molecule is infinitely fast as compared with the rate of precursor formation on an active site. Mathematical relations are derived for the active site coverage or coke content and for the number of sites rendered inaccessible by coke molecules blocking the pores. The relations coke content vs. time and deactivation (or activity) function vs. time are derived for single pores open on one end and on both ends, for a set of parallel pores and for various networks of interconnecting pores. The application of the theory to the kinetic analysis of a catalytic process subject to catalyst deactivation by coking and to the design of a tubular reactor is; illustrated.

1. Introduction Many reactions of the petroleum and petrochemical industry are accompanied by coke-forming side reactions. Butt (1972, 1978) thoroughly reviewed the literature on the coking and deactivation problem. In a recent paper, Froment (1976) developed a theory relating the deactivation of the catalyst to its coke content. To account for the deactivation, the rate of the main reaction A B was written +

rA

= rAoqA

05

(PA

5 1

(1)

with the initial rate in the absence of coke, rAo,given by 0019-7874/79/1018-0245$01 .OO/O

FA'

= kAC?fi ( cA, c~i~K.4 ,KB)

(2)

and the deactivation function for the main reaction, pA, by

..=(c-> c, - CCl

nA

(3)

nAequals one for a single site main reaction and two for a dual site main reaction. C, is the total concentration of active sites and Ccl is the concentration of sites covered with coke or coke precursor. p A accounts for the decrease in the number of active sites available for the main reaction

0 1979 American Chemical Society

246


because of the coverage with coke or coke precursor. As is clear from (3), for a single site mechanism it is the fraction of sites remaining active and varies between one and zero. Notice that cpA could also be called an “activity function”. In general the coverage of active sites by coke also deactivates the coke formation itself, although not necessarily in the same way as the main reaction, so that with and

Equations 4, 5, and 6 may also be written

with k, = k,‘Mc, where M, is the molecular weight of the coke “molecule”. So far only empirical relations have been used for the deactivation function. Several authors have related cp to time, some of them by means of a simple exponential function, although time is not the only variable in the deactivation (Szepe and Levenspiel, 1971; Wojciechowski, 1968; Weekman and Nace, 1970). In a recent review, Froment (1976) has given several reasons for relating cp to the coke content, C,, of the catalyst. De Pauw and Froment (1975) and Dumez and Froment (1976) experimentally observed an exponential relation between cp and C, in n-pentane isomerization on a platinum reforming catalyst and in 1-butene dehydrogenation on a chromia-alumina catalyst. The present work investigates one mechanism that could explain this type of functional relation, namely that of site coverage and pore blockage by coke growing from the active site where it was deposited. 2. Basic Assumptions and Theory of Deactivation by Active Site Coverage and Pore Blockage The following assumptions are made in this work. (a) The main reaction and the coking reaction occur by a single site mechanism on the same active sites (nA = n, = 1). (b) The growth of the coke molecule is fast with respect to the rate of formation of the coke precursor on the active site. The size reached by the coke molecule depends upon the reaction conditions. The molecule grows to such an extent that it may block the pore. (c) The coke precursor deposits in a random way on the active sites. (d) There is no diffusional limitation on the rate of the main reaction, so that the concentrations of the reaction partners are uniform inside the pores of the catalyst particle. Assumptions a and b imply that the deactivation functions for the coking and the main reaction are identical. Consider now, by way of example, a single ended pore and let the reaction conditions be such that a coke molecule reaches such dimensions that the pore is blocked. The coke molecule, although it covers only one site, will in general render more than one site inaccessible to the reactants. The deactivation function cpA = cpc = cp is still defined as the fraction of sites remaining active for nA = n, = 1but it no longer equals (C, - Ccl)/C,. A distinction has to be made between the number of sites covered by coke and the number of sites deactivated, i.e. covered and/or inaccessible. To consider deactivation by both coverage and blockage, an approach based upon probability

theory was developed. Most of the equations derived in the present work are based on the two following formulas of probability theory P(AB) = P(A)P(B/A) (8)

P(A

+ B) = P(A) + P(B)

P(AB) (9) In words: the probability that events A and B occur simultaneously, P(AB), is the product of the probability that A occurs and of the probability that B occurs, provided that A has occurred. The probability for A and/or B to occur, P ( A + B), is the sum of the probabilities that A and B occur, P(A) + P(B), minus the probability that A and B occur simultaneously. -

To start with, however, the problem of deactivation by site coverage alone, dealt with in the Introduction, will be rephrased in terms of probabilities. Let S ( t ) be the probability that a site is still active at time t , in other words not covered (and not blocked, of course). S ( t ) is nothing but the fraction of sites remaining active, i.e. cp, defined above. Equation 7 may then be written dS _ dt and after integration from 0 to t

Let r>dt be the probability that a site is covered in the time interval dt. From (8) the probability S ( t + dt) that a site is still active at t dt is the product of the probability that the site is active at time t and of the probability that the site is not covered in the time interval dt, so that

+

and

rso = 1d S S dt

Notice that r:, the initial rate of fractional site coverage, is proportional to the initial rate of coking r,O and that, when the gas phase conditions are kept constant

S ( t ) = exp(-r,0t) (14) Consider now the case of coverage and blockage. The probability that a site is still active, in other words, not covered and not blocked, is still represented by cp. This probability is the product of two probabilities. The first P ( t ) ,is the probability that the site is accessible at time t , the second S ( t )is the conditional probability that the site is not covered, provided that it is accessible a t time t , so that cp(t) = P(t)S(t) 0 IP ( t ) I1 (15) Evidently cp (coverage + blockage) C cp (coverage). When there are no concentration or temperature gradients in the pore, S ( t ) has the same value for all the sites along the pore. At the pore mouth itself P(t) = 1, but P(t) decreases from the mouth to the end of the pore. Consequently, cp depends on the location and is therefore called the “local deactivation function”. After integration over a pore it is written +, after integration over a network of


pores, i.e., over a particle, 9. Let w ( t ) be the probability that a given site is covered with coke at time t. The probability that the site is covered a t t dt equals the probability that the site is covered at time t plus the probability that the site is covered during the time interval dt, provided that the site was not covered and not blocked at time t. In mathematical terms

0 0

.r;tO

.

0

c

d

N=10 S

+

247

0

from which w =

Jtr:qdt

w is also a local value. For the average value of w over a single pore the symbol i3 will be used, for the integral value over a whole particle C!. The average degree of coverage, 0,is related to the experimental coke content C, by C, = CtMcC! (17) The relation 9 vs. C , therefore follows directly from the relation 9 vs. C! for a particle or vs. i3 for single pores. This relation will now be established for pores with a single opening, for pores with both sides open, and for different types of networks of pores. 3. Single-Ended Pores with a Deterministic Distribution of Active Sites Let the sites be numbered from 1, a t the pore mouth, to N,, at the end of the pore. Then, since site 1is accessible a t any time, P(1,t)= 1. From (€9,the probability that site n is accessible a t time t is given by P(n,t) = P(n - l , t ) S ( t ) (18) and from (14) P(n,t) = P(n - 1,t) exp(-r,0t)

S I l E NUMBER N

Figure 1. Local deactivation function vs. site number for a single ended pore with a deterministic distribution of sites. 0 0

0 0

0

m

0

(D 0

0

3 0 0

so that

P(n,t) = exp(-(n - l)r,0t) Further, from (15) it follows that p ( n , t ) = exp(-nr,0t)

N 0

(19) (20)

0

B lr;t. oi

and over the whole pore

0

0

i,. 0

1

i

i

i

5

i

i

S I l E NUMBER N

Figure 2. Local degree of coverage vs. site number for a single ended pore.

The probability that a given site is covered with coke follows from (16) 1 1 w(n,t) = - - - exp(-nr,0t) (22) n n For the entire pore the degree of coverage is given by

;(t) =

-

, N. 1 -'I - C -[1 - exp(-nr,Ot)]

(23) Nsn=ln Figure 1 represents the value of the deactivation function cp (n,t)vs. the site number for a pore having 10 sites and with the dimensionless time, r'gOt, as a parameter. It should be stressed that the sites are not necessarily equidistant, so that the curves of the figure are not exactly profiies with respect to the distance inside the pore. Figure 2 shows the coverage probability w(n,t)of the different sites for a pore with N , = 10 and again with r:t as a parameter. Figures 3 and 4 show act) and i3(t), respectively, as a function of reduced time r;t, with the total number of sites N , as a parameter. Figure 5 shows the relation between the deactivation function for the pore act) and the degree of coverage, &(t)which is proportional to the coke content,

for a number of sites in the pore ranging from 1 to 10. 4. Single Ended Pore with a Stochastic Distribution of Active Sites Let adx be the probability of occurrence of an active site in the interval dx and let L be the total length of the pore. Evidently, PL is the average number of sites in such a pore. Then, again from (8) P(x + dx, t ) = P(x,t)(l - a dx(1 - exp(-r:t))) =

P(x,t) + ( 5 ) d x

so that the probability that a site is accessible is given by

P(x,t) = exp(-ax(l - exp(-r,0t))) and since S ( t ) = exp(-r:t) and using (15) cp(x,t) = exp(-r?t) exp(-ax(1 - exp(-r:t)))

(24) (25)

1 exp(-r'gOt) +(t)= (1- exp(-aL(l - exp(-r,0t)))) aL (1- exp(-r?t))

248


1

w ( x , t ) = -(1 ax

-

exp(-ux(1 - exp(-r;t))))

(27)

a(t) = exp (-rSot)

(1 - exp(-aL(1

-

exp(-r,0t))))dt (28)

cp(x,t) vs. x/L, w ( x , t ) vs. x/L, and +(t),& ( t )vs. r',Ot as well as vs. 3 are almost identical with those of Figures 1-5 with N,= 10 replaced by uL = 10.

+

5. Pores Open on Both Sides with a Deterministic Distribution of Active Sites The probability P(n,t) that site n is accessible is calculated from eq 9 with P ( A ) the probability that site n is accessible from one side of the pore and P(B) the probability that site n is accessible from the opposite side of the pore.

P(n,t) = exp(-(n - l)r,0t) + exp(-(N,

- n)r,0t) exp(-(N, - 1)r:t)

S ( t ) = exp(-r,0t) cp(n,t)= exp(-nr,0t)

+ exp(-(N,

-n

(29)

-

exp(-(N, - n

1 + 1)rsot))- -(1

-

exp(-N,r,0t)) (33)

1 -(1

-

exp(-N,r>t)) (34)

N,

N,

The deactivation and coverage profiles with respect to the site number reflect the symmetry in the pore, while the vs. r2t and 3 vs. r,0t curves are analogous to those in Figures 3 and 4. An inflection point develops in the curves @ vs. 3 for all values of N,, except N,= 1, of course.

+

6. Pore Open on Both Sides with a Stochastic Distribution of Sites Along the Pore The following formulas can easily be derived. - exp(-r,0t))) + exp(-u(L - x) x (1 - exp(-r,0t))) - exp(-aL(I - exp(-r,0t))) (35)

P(x,t) = exp(-ux(1

S(x,t) = exp(-rS0t)

(p(n,t)= exp(-r,0t)[exp(-ax(l

(36)

+

- exp(-r,0t))) exp(-u(L - x ) ( l - exp(-r,0t))) - exp(-uL(1 ex~(-r,0t)))l (37)

(1 - exp(-r,0t))

X

aL and again figures analogous to Figures 1-5 are obtained. 7. Set of Independent Pores with a Deterministic Distribution of Sites A catalyst particle is considered to contain a set of noninterconnecting pores with different diameters. The pores are grouped according to their diameter into a number of classes k. A fraction of the pores has diameters exceeding the size D, of the coke molecule so that they cannot be blocked. Let Do') be the diameter of a pore of class j , go') the number of pores pertaining to class j , and N,G)the total number of active sites in a single pore of class j . Let +G,t) represent the deactivation function for a pore of class j and 3G,t) its coverage. If 1 is a number such that D(1) = D, the global deactivation function 9 for the set, i.e., the whole particle can be written

[ CgCi)N,o')+O',t)+ 1

9=

exp(-N,r',Ot) (31)

exp(-rS0t)

(1 - exp(-aL(I - exp(-rS0f))))dt1 -(1 - exp(-aL(1 - exp(-r>t)))) (40)

(30)

+ 1)r;t)

'

7: 1

a(t) = -

j=1

k

exp(-r,0t) J=1+1 C gWN,W]

/ hAN,O.) j=l

(41)

In (41) the denominator is nothing but the total number of sites in the entire set; the first term in the numerator is the number of sites remaining active in pores with diameter D ID,, in which coverage and blockage occurs and the second term gives the number of sites not deactivated by coverage in pores with diameter D > D,, in which blockage cannot occur. The global coverage 0 for the whole particle is written

k

1

i k

In (42) the first term in the numerator represents the number of sites covered in pores with diameter D i D,, the second term the number of sites covered in pores with diameter D > D,. If the set consists of pores open on one side only, +G,t) and 3O',t)are given by (21) and (23). For pores open at both sides they are given by (32) and (34). From an inspection of (41) and (21) or (32) it is clear that the relation between 9 and time is not as simple as has been postulated in previous empirical work. By way of example, a set of pores is chosen here which is suggested by the work of Dumez and Froment (1976) on the dehydrogenation of l-butene into butadiene on chromia alumina. This catalyst has a considerable fraction of pores with diameters between 25 and 50 A. From the work of Eberly (1966) and Levinter (1967) it is evident that coke molecules on catalysts can easily exceed such a size. The total pore volume of the catalyst was 0.311 cm3/g. The pore size distribution was determined by nitrogen adsorption and mercury penetration and is given in column 2 of Table I in terms of pore volume for the pores belonging to the classes of pores with an average diameter given in the first column. Assuming the pores to be cylindrical enables the total length corresponding to each average diameter to be calculated. The results are reported in column 3. From the length and the diameter, the surface area corresponding to each diameter is computed and column 4 reports the fraction of the total surface area


, "0.00

b

\\\

b>-l+-_j; 1.00

2.00

e'

3.00

Ll.00

5.02

Table I. Pore Size Distribution of a Chromia-Alumina Catalyst for Butene Dehydrogenation (Dumez and Froment, 1976)

vol, cm3/g

25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 175 180 185 190 195 200 300 400 500 800 1500 2500 3500 4500 7500 15000 25000

0.002 0.008 0.01 0 0.008 0.003 0.002 0.003 0.005 0.009 0.013 0.011 0.01 1 0.008 0.008 0.006 0.008 0.007 0.007 0.008 0.006 0.007 0.005 0.003 0.001 0.001 0.002 0.002 0.002 0.003 0.002 0.002 0.009 0.01 3 0.007 0.011 0.01 2 0.009 0.009 0.007 0.010 0.030 0.022

total length, m/g 4.0 x 1.1x 1.0x 6.4 x 1.9 x 1.0 x 1.3 x 1.8 x 2.7 X 3.4 x 2.5 x 2.2 x 1.4 X 1.3 x 8.5 x 1.0 x 8.1 x 7.4 x 7.7 x 5.3 x 5.7 x 3.8 x 2.1 x 6.5 x 6.1 x 8.3 x 7.9 x 7.4 x 1.1 x 6.7 x 6.4 X 1.3 x 1.0 x 3.6 x 2.2 x 6.8 x 1.8 x 9.4 x 4.4 x 2.3 x 1.7 x 4.5 x

0.00

1.00

3.00

2.00

5.03

11.00

rg t

Figure 3. Global deactivation function vs. reduced time for a single ended pore.

av diam, A

249

lo8 109 109

los

10s 108

lo8 lo8 lo8 108

lo8 108

lo8 lo8 107 108 10' 10' 10' 10' 10' 107 10'

lo6 lo6 lo6 lo6 106 10'

lo6 lo6 107 107

lo6 106

lo5 105 104 104 104

lo4 103

Figure. 4. Global degree of coverage vs. reduced time for a single ended pore. 0 0

0 0

fraction of total surf. area 0.03134 0.10380 0.11163 0.07834 0.02546 0.01567 0.02154 0.03231 0.05386 0.07 246 0.05778 0.05386 0.0372 1 0.03427 0.02448 0.03134 0.02546 0.02448 0.02742 0.01958 0.02154 0.01 469 0.00881 0.00294 0.00294 0.00392 0.00392 0.00392 0.00588 0.00392 0.00392 0.01750 0.01273 0.00490 0.00490 0.00294 0.00098 0.00098 0.00059 0.00049 0.00078 0.00029

corresponding to each of the pore classes. The number of pores of each class is given in column 3 of Table 11. It is proportional to the total length of the pores of that class.

Figure 5. Global deactivation function vs. global degree of coverage for single ended pores. Table 11. Site Distribution and Pore Size Distribution Used in the Simulation of the Deactivation b y Site Coverage and Pore B l o c k a e diam, no. of no. of no. of A sites pores sites/pore 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 > 100

32 99 110 78 28 15 21 32 54 72 57 52 36 34 24 31 225

4 11 10 6 2 1 1 2 3 3 3 2 2 2 1 1

8 9 11 13 14 15 21 16 18 24 19 26 18 17 24 31

From electron microscopy observations the length of a pore was taken to be 5 pm. Consequently, the number of pores

250

Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979 Nr3

-.

0 e

El

-

_. D

m 0

-

-.

(D 0

0

-

-.

3 0 0

-

E2

-.

N 0

0

0

oioo

0

0

0120

0140

2 0

’

>I30

1.08’

Figure 7. Schematic representation of type 1 network.

proach developed here may not be fulfilled in butene dehydrogenation.

8. Networks of Pores In general, the physical and chemical phenomena inside catalysts are not modeled in terms of the fine structure. In some cases, however, phenomenological properties like effective diffusivity have been explained and predicted by means of models for the structure of the pore network inside the catalyst (Mingle and Smith, 1961). Carberry (1962) and before him Wheeler (1951) accounted explicitly for the pore structure in dealing with selectivity problems for consecutive reactions. Pore structures have been dealt with in a stochastic way in other areas by Broadbent and Hammersley (1957) and Pismen (1972, 1973). In the present paper this stochastic approach will be applied to the catalyst deactivation problem. It extends recent work by Beeckman, Froment, and Pismen (1978) by enabling the coke profiles inside the pores of the catalyst to be described. (a) Type 1 Network. In this type, represented in Figure 7, all the pores have the same radius and the pores do not end in the catalyst particle itself. The branching of pores and the location of sites along the pores are random. Consider the pore 1-11which is branching in I1 into two different pores 11-111and 11-IV, thus yielding two exits. From I11 and IV onwards the branching continues, yielding two sub-networks with sets of exits E, and E,, respectively. Let PN be the probability that I1 is accessible from the set of exits generated from the Nth branching generation and let PN’(L1,L,) be that probability when the pores 11-111 and 11-IV have a length between (L1,Ll+ dL1) and (Lz,L2 + a,), respectively. It is clear from Figure 7 that I1 can be reached both from the set of exits El and E2. Then from (9) PN’(L1,Lz) = P(E1- 11) + P(E2 11) - P(E1II)P(EP 11) (43)

-

-

-

-

P(E, 11) and P(E2 11) can be calculated from (8) noticing that, since each subnetwork contains the same number of branching generations, N - 1, that P(E1 111) = P(E2 IV) = while P(II1 11) and P(IV 11) are calculated from the single pore formulas. Equation 43 then becomes

-

PN’(W2)

-

--

=

PN-l[(exp(-uL1(l - exp(-r:t))) + exp(-u12(l exp(-r,0t))) - PN-l exp(-u(L1 + L2)(1 - exp(-r,0t))))I (44)

Let v d L be the probability of branching in the interval (L,L + The probability that a pore has a length between

a).


251

L and L + dL is then u exp(-uL)dL. Using eq 8 and 9, PN can be related to PN'(Ll,L2) by

PN=

S,-dL S,

-u2

1

exp (-vLl) exp (-uL2)P,v'( L ,L2)dL2 (45)

Substitution of (44) into (45) and integration leads to the recursion formula PN

I

(46)

= 2P'PN-l - P 2 P 2 N - 1

3

DOL00

L

01051 0 1905 02231

5 6

and evidently Po = 1, where

i

or

t

D

0 0

6

(47)

P is nothing but the probability of getting through a randomly taken pore of the network. The probability that node I1 is accessible from the exits a t the surface of a particle containing a network of pores with a large number of branching generations is given by

Pa = limN

-

,P N = 2

PP,

-

P2Pm2

0 D

0

c

i

0

(48)

The solutions of the quadratic (48) are

0

(D 0

o

Pa =

2P- 1

P2

(49)

I

O

-6

2

00202 OOLO0

1

w

----Ir

Pa = 0

(50)

Since Pa is constrained between 0 and 1 and since Pa(P = 1) = 1 the solution (49) has to be selected when 0.5 5 P 5 1 and the solution (50) when 0 I P I 0.5. Substitution of (47) into (49) yields for Pa

rz t -

m

o

=?

L

5 b

01051 01905

02231

0

0

0

"0.00

0.20

0.qo

0.60

0.80

1.03

XIL

The local deactivation function in a position x in a pore of length L at a time t is given by

(a(L,x,t)= exp(-r,0t)(P,(exp(-ax(l - exp(-r;t))) + exp(-a(L - x ) ( l - exp(-rS0t))) -P, exp(-aL(l exp (-r;t)) )) 1 (52) The deactivation function for a pore of length L is given by

exp(-uL) uL+(L,t)dL

@ ( t )=

(531

For the coverage in the network L m v exp(-uL)aLO(L,t)dL

L-u

Carrying out the integrations indicated in (53) and (54)

(54) exp(-vL)uLdL

-

%Y

-

3

(1 - e~p(-r;t))~ (56)

Figures 8 and 9 show cp(L,x,t) and w(L,x,t) vs. x/L for for L = l / u , which is the mean pore length in the network. The parameter is the dimensionless time r2t. Figure 10 shows the relation between the deactivation function @ and the global coverage 3, which is proportional to the coke content, for various values of u/v,which is the mean number of sites in a pore. These curves do not have the shape observed by Dumez and Froment (1976) and De Pauw and Froment (1975). (b) Type 2 Network. This type is represented in Figure 11. It differs from Type 1 because some pores end inside the catalyst particle itself. a j u = 1 and

exp(-uL)aLdL

O(t) =

where O(L,t)is the degree of coverage for a pore of length L, given by

3 ( t ) = 1 - exp(-r;t)

and for the whole network

s,

Figure 9. Local degree of coverage in a pore of length 1 / u for a type 1 network.


252

inaccessible even at zero time. This-happensbecause it is possible that in a particular generation of pores, say N = 2 in Figure 11, all the pores terminate in dead-ends and thus it is impossible to get into or out of this subsystem of pores. This fraction is obviously not accounted for in the expressions for the deactivation function and the degree of coverage. The local deactivation function in a position x in a pore of length L and at time t is given by

0

+ exp(-a(L - X)(I

exp(-ax(l- exp(-r:t)))

PmU

exp(-r:t)))

-

exp(-aL(l V + T

- exp(-r:t)))

-

1

(60)

For the whole network the deactivation function and the degree of coverage are given by Figure 10. Global deactivation function vs. global degree of coverage for a type 1 network.

W) =

xm(u

+

7)

exp(-(v

N=3

+ r ) ~ ) a ~ ; . ( ~ , t )0 d ~ +/ l ~ ( v exp(-(v + 7)L)aLdL (61) T)

El

7)

exp(-(u

+ ~ ) L ) a i d L(62)

Carrying out the integration of eq 61 and 62 yields

/

/‘’

(63)

/y # E X l T S

Ep

Q ( t )= 1 - exp(-r:t)

-

//

Figure 11. Schematic representation of a type 2 network.

Let r dL be the probability that a pore ends in an interval dL. It can easily be shown that the probability for a pore to have a length between L and L + dL and to branch or end is given by v exp(-(v + 7)dL or 7 exp(-(v 7)L)dL respectively. With this in mind and using eq 8 and 9 the equation for P N is easily arrived at by following the reasoning given in detail for type 1 network.

+

PN= J,mdLIJm exp(-(v 0

+

7)(&

+ L,)) X

(vr exp(-aLl(l - exp(-r,Ot)))PN_l+ exp(-aL2(l- exp(-r,0t))) + v2[exp(-aLl(l exp(-r,Ot)))PN-l + exp(-a12(l - exp(-r,Ot)))PN-lP’N-Iexp(-a(Ll + LJ(1 - exp(-rs0t))))l1dL2(57) Integration of eq 57 leads to the recursion formula P N = 2PPN-1 - P 2 P 2 N _ 1 (58) PN-1

with

P = l m v exp(-(v 0

+ 7)L) exp(-aL(l

-

exp(-r:t)))dL

P, is derived in the same way as for type 1 network

In this case Pm(t= 0) = 1 - ( 7 / ~ # ) ~0. This means that ) ~the sites is in this type of network a fraction ( 7 / ~ of

1-

(7/V)2

The curve @ vs. Q is very similar to the one shown in Figure 10. (c) Type 3 Network. This type differs from type 1 by the distribution of pore diameters. Inside the pores with diameters exceeding D, no blockage is possible. Define u(D)dx as the probability of occurrence of an active site in the interval dx of a pore with diameter D, u(D)dL the probability that a pore with diameter D branches in the interval dL, and P(D,D1,D2)dDldD2the probability that a branching pore of diameter D gives rise to pores with the respective diameters (D1,D1 + dDl)and (D2,D2 dD2). Again as for a type 1 network first PN’(D,Dl,D2,L1,L2)is calculated. Then through (8) and (9) PN’is related to PN-, and finally the probability that a site is accessible from the surface is obtained after a quadruple integration.

+


253

the micropores can only branch into two macropores and that the macropores can branch into two macropores or into a micropore and a macropore. Then, from the definition of type 4 itself, it follows that P(DJ = P(D,) = 1. This means that macropores can in no way get blocked, while micropores can only block in the micropores themselves. Further, let y be the probability that a site is located in a micropore. Obviously y equals the fraction of the catalyst surface located in the micropores. From. (37) the local deactivation function q(L,x,t) for a site located in a micropore is given by ql(L,x,t)= exp(-r,0t)(exp(-ux(1 - exp(-r,0t))) exp(-a(L - x)(l - exp(-r,0t))) - exp(-aL(l exp(-r,0t)))l (69) The global deactivation function and degree of coverage for the micropores are given by

+

G1(t) = i m 0 u exp(-uL)aLB,(L,t)dL/imu n exp(-uL)aLdL

(70) Figure 12. Global deactivation function vs. global degree of coverage for a type 3 network. (64) leads to an expression identical with (46) of type 1, but with

Finally, the probability P, for a node to be accessible from the external surface of a catalyst particle with a large number of branching generations in the pore network is given by

Integration of (70) and (71) and accounting for the deactivation function and degree of coverage of the macropores, the deactivation function and degree of coverage for the whole particle is given by 9 ( t )=

y exp(-r:t)(

1-

[

:(I

-

exp(-r?t))

1 + -(1 CT U - exp(-r,0t))

I;)+

The local deactivation function at a position x for a pore of length L with a diameter D a t time t is given by: in pores with D ID,

+

dD,L,t,x) = exp(-r,0t)P,(exp(-a(D)x( 1 - exp(-r,0t))) exp(-a(D)(L - x)(l - exp(-r,0t))) - P, exp(-o(D)L(l exp(-r,0t)))l (66) in pores with D > D, p(D,L,t,x) = exp(-r,0t)(2Pm- Pm2)

(67)

The deactivation function and the degree of coverage for the whole particle are given by

+(t)=

& & mg(D)u(D)exp(-v(D)L)u(D)LB(D,L,t)dDdL / Lm&,g(D)v(D) exp(-v(D)L)a(D)LdDdL (68)

Q ( t )= &m&mg(D)v(D)exp(-u(D)L)u(D);(D,L,t)

X

dDdL/ &m&mg(D)v(D) exp(-v(D)L)u(D)LdDdL Figure 12 gives @ vs. R for ( ~ / v )= 2 which is the mean number of sites per random pore and for several values of D,. The curves still do not have the observed roughly exponential shape. (a) Type 4 Network. Type 4 is a micro-macro pore network and in fact a simplified Wheeler model. Let D1 be the diameter of the micropores and D2 the diameter of the macropores while D1 < D, and D, > D,. Assume that

The first term of (72) and (73) takes into account the deactivation and coverage in the micropores, the second term that in the macropores. Figure 13 gives 9 vs. Q for y = 0.8 and for several values of the parameter u/v. Figures 14 and 15 give 9 and R as a function of r;t, for y = 0.8 and for the same values of a / u as in Figure 13. The value of 0.8 for y was chosen to enable a comparison with the curves derived in section 7 for the chromia-alumina catalyst used by Dumez and Froment (1976). It was mentioned in section 7 that the micropores of that catalyst have an average length of 5 pm; accordingly the parameter u then equals 1/6 pm. The average number of sites per micropore in Figure 13 ranges from 0.5 to 10. 9. Kinetic Analysis of Deactivating Systems The various formulas given above for the coke content and for the deactivation function form a fundamental basis for the kinetic analysis of deactivating systems, provided that the basic assumptions mentioned in section 2 are fulfilled, of course. Suppose a single site main reaction, with the reference component represented by A and one (lumped) coking reaction. A certain number of rival kinetic models for the main and for the coking reaction have to be confronted with the experiments. The coke content is preferably followed as a function of time in an electrobalance type equipment operated in a differential manner (Froment, 1976). The data treatment of the experimental


254

0

0

0

0 0

parameter

0 0

a

// /

s

i

!

! 0.00

1.00

3.00

2.00

R

u.00

5.03

rz t

Figure 13. Global deactivation function vs. global degree of coverage for a type 4 network.

.-

Figure 15. Global degree of coverage vs. reduced time for a type 4 network.

0

0 0

r;

parameter 4

I

V

t

m 0 0

m 0 0

@

rg t

Figure 14. Global deactivation function vs. reduced time for a type 4 network.

results for rA and C, should preferably be performed simultaneously. For this purpose an objective function, 3 , is defined 3 = w1 C ( r A - FA)' + w2 -

c(c, e,.)'

11

12

w1 and w 2 are weighting factors while l1 and 1, are the ?umber of data points on rA and c,, respectively. ?A and C, are estimates for rA and C, based on the kinetic models to be tested. ?A = rAo@(t ) where rAois a function of concentrations, containing kiqetic and adsorption parameters, given by (21, whereas C, is related to R(t) through eq 17. The expressions for 9(t)and R(t) depend upon the model for the porous structure of the catalyst. The parameters associated with the pore structure should be determined as much as possible from independent physical observations. Suppose, by way of example, that the catalyst has a fine structure according to type 4 network. In that case 9 is given by (72) and R by (73). 9 and R contain y,a/v and the kinetic parameters

comprised in r:. The values of y, the fraction of the catalyst surface area located in the micropores, and l/v, the mean length of the micropores, are determined independently from adsorption measurements and electron microscopy observations respectively. The site density u in the micropores is related to the total concentration of sites C , by a = "D,C,NA/SA where D , is the diameter of the micropores and NA is the Avogadro number. The kinetic equation r,O contains the concentrations of the reaction partners and their adsorption constants which also appear in the rate equation for the main reaction (2). Finally, the relation between C, and 52 (17) requires the evaluation of M,, the molecular weight of the coke molecule. The simultaneous analysis of the rA and C , data through the objective function defined above then yields estimates for the model parameters: Ct, kA, k,, KA, K g , and eventually M,. It is obvious that in order to evaluate significantly all model parameters, data are necessary both on the main reaction and the coke content although these need not be determined simultaneously. The discrimination between rival kinetic models then further proceeds along classical ways (Froment, 1975). Data on the coke content profile in tubular reactors may give precious indications as to the mechanism of coking. This will become clear from the next section. 10. Design of a Tubular Reactor Subject to Coke Deposition Consider an isothermal tubular reactor with plug flow in which a reversible reaction A B is taking place. Let the catalyst have a pore network of type 4, i.e. containing macro- and micropores, but let there be no internal or external limitation to mass transfer. When the surface reaction on a single site is rate controlling, the rate of reaction in the absence of coke is written

(CA-:) rAo = k K C A

A

+

t ( l KACA

+ KgCg)

(74)

The continuity equation for A may be written (75)

Ind. Eng. Chern. Fundam., Vol. 18, No. 3, 1979

ij

Ij a

4 = 5

yz0.e

0

V

0

y.0.e

21

255

Q=5

V

1

0

1

0

2

0 83

2

3 L

148 2 23 3 L5 6 85

0 26 0 59

5 5

3

R 3 D

n

116 248 5 26 L5

4 5

N "

a

0 0

oOIOO

D

a

1

'

0120

'

O!UO

'

OlSO

'

0180

'

llO?

uo

x/ L

Figure 16. Global degree of coverage vs. axial distance in a tubular reactor with parallel coking.

When the deactivation is relatively slow the reactor may be considered to operate in the quasisteady state and the first term in (75) may be dropped. The rate of change of coke on the catalyst follows from

cc

-=@ :r at

or

Following the procedure and nomenclature of Froment and Bischoff (1961), the coke is considered to be formed either from A (parallel coking) or from the product B (consecutive coking). In the first case, for an irreversible single site coking reaction (77) in the second case

where r: is now a function of position in the reactor, through the local values of CA and CB. I t can be shown that for variable gas phase conditions eq 72 for the global deactivation function in a type 4 network becomes

and, since in a tubular reactor r: is not a constant any more (viz. (10))

The following parameter values were used in the calculations: y = 0.8, o/v = 5, and KA = KB = 50 m3/kmol. Figure 16 shows D (or coke content) profiles in the reactor for parallel coking at finite times. The coke profile is decreasing as predicted earlier by Froment and Bischoff (1961, 1962) by means of an analytical approach restricted to first-order reactions and with an empirical, exponential deactivation function expressed in terms of the coke content. For consecutive coking, given in Figure 17, the coke profiles originate from zero and are ascending at finite

o!so

I

oleo '

i,o$

x/L Figure 17. Global degree of coverage vs. axial distance in a tubular reactor with consecutive coking.

times, again as predicted by Froment and Bischoff. 11. Conclusion When the deactivation by coke deposition not only occurs by site coverage but also by pore blockage, the detailed internal structure of the catalyst has to be accounted for. The approach developed in this paper permits the rigorous derivation of the deactivation function for a wide variety of pore structures. When the catalyst fine structure has been characterized by means of the usual physical methods, the formulas presented here allow truly fundamental coking parameters to be determined. This is what is required to come to a better understanding of the coking phenomenon, as a first step toward the tailoring of catalysts less active in coke formation and less sensitive to the coke content. If coking parameters are determined from integral tubular reactor studies it is evident that the average coke content does not provide sufficient information. What is required to get an insight into the coking mechanism is the actual coke profile. In a subsequent step the rate parameters can then be estimated by fitting of the data. Acknowledgment J. W. Beeckman is grateful to the 1.W.O.N.L.-I.R.S.I.A. for a research fellowship over the period 1975-1978. The authors are indebted to Professor L. Pismen (Technion, Haifa, Israel) for valuable discussions on the characterization of random pore structures. Nomenclature CA,CB = concentration of reagent and product, kmol/m3 C, = coke content of the catalyst, kg of coke/kg of cat. Ccl = concentration of sites covered with coke, kmol/kg of cat. C, = total concentration of sites, kmol/kg of cat. D = diameter of a pore, D, = size of a coke molecule, A fl, fz = functions determined by the main reaction mechanism and the coking mechanism, respectively k = number of classes in a set of pores hA = rate coefficient for the main reaction, (l/h)(kmol/kgof cat.) l - n ~ k , = rate coefficient for the coking reaction, kg of coke/kmol"c kgl-nc cat. h h,' = rate coefficient for formation of coke precursor, kmol/kg of cat.)'-"c/h KA,KB = adsorption constants, m3/kmol L = length of a pore, m Mc = molecular weight of a coke molecule

a

256


N = number of branching generations N , = total number of sites in a single pore n = site number in a single pore nA = power of total concentration of sites in the coking rate equation n, = power of total concentration of sites in the coking rate equation N A = Avogadro number, l/kmol P = accessibility function PN = accessibility of a node from the exits of N branching generations P = mean probability of getting through a pore of a network r.4 = rate of the main reaction, kmol/(kg of cat. h) rAo = rate of the main reaction in the absence of coke, kmol/(kg of cat. h) rc = rate of coking, kg of coke/(kg of cat. h) r: = rate of coking in the absence of coke on the catalyst, kg of coke/(kg of cat. h) r: = initial rate of fractional site coverage, l / h SA = surface area of the catalyst, m2/kg of cat. S = probability that an accessible site is active t = time, h u = superficial gas velocity, m/h x = axial coordinate in a pore Greek Letters ~(D,D1,D2)dDldD2 = probability that a branching pore of diameter D branches in pores of diameters D1 and D2 y = probability that a site is located in a micropore v d L = probability that a pore branches in a length interval

dL

= bed density, kg of cat./m3 of reactor udx = probability that a site is located in a length interval dx

r d L = probability that a pore ends in an interval dL = local deactivation function @ = deactivation function for a single pore = deactivation function for a network of pores, Le., for a catalyst particle w = local degree of coverage 0 = degree of coverage in a single pore Q = degree of coverage in a network of pores, i.e., for a catalyst particle cp

L i t e r a t u r e Cited Beeckman, J. W.; Froment, G. F.; Pismen, L. Chem. Ing. Tech. 1976, 50, 960. Butt, J. B. Adv. Chem. Ser. 1972, No. 109, 259. Butt, J. B. International Symposium of Chemical Reaction Engineering, Houston, March 1978. Broadbent, S. R.; Harnmersley, J. M. R o c . Cambridge Phil. SOC. 1957, 53, 629. Carberry, J. J. Chem. Eng. Sci. 1962, 17, 675. De Pauw, R.; Froment, G. F. Chem. Eng. Sci. 1975, 3 0 , 789. Dumez, F. Ph.D. Thesis, Rijksunlversiteit Gent, 1975. Dumez, F.; Froment, G. F. I d . Eng. Chem. Process Des. Dev. 1976, 15, 291. Eberley, P. E.; Kimberlin, C. N.; Miller, W. H.; Drushel, H. V. Ind. Eng. Chem. Process Des. Dev. 1966, 5 , 2. Froment, G. F.; Bischoff, K. B. Chem. Eng. Sci. 1961, 16, 189. Froment, G. F.; Bischoff, K. 8. Chem. Eng. Sci. 1962, 17, 105. Froment, G. F. AIChE J. 1975 21, 1041. Froment, G. F. Proc. Sixth Int. Congr. Catal. 1976, 1, 10. Levinter, M. E.: Panchenkov, G. M.; Tanatarov, M. A. Int. Chem. Eng. 1967, 7, 23. Mingle, J. 0.; Smith, J. M. AIChE J . 1961, 7, 243. Pismen, L. M. Dokl. Chem. Techno/. 1972, 207, 238; 1973, 211, 126. Szepe, S.; Levenspiel, 0."Proceedings,4th European Symposium on Chemical Reaction Engineering", Pergamon Press: London, 1971, p 265. Weekrnan, V. W.; Nace, D. M. AIChE J., 1970, 16, 397. Wheeler, A. Adv. Catal. 1951, 3 , 249. Wojciechowski, B. W. Can. J. Chern. Eng. 1966, 46, 40.

p~

Received for review July 18, 1978 Accepted February 15, 1979

Residence-Time-Distribution Studies on a Gas-Liquid Countercurrent Packed Column with Intermittent Voids along the Axis Pradeepchandra Trasl and Soon-Jai Khang" Chemical and Nuclear Engineering Department, University of Cincinnati, Cincinnati, Ohio 4522 1

Residence-timedistribution (RTD) studies were made on a gas-liquid countercurrent packed column with cylindrical voids placed intermittently along the axis of the column. The liquid flow behavior of this packing arrangement was compared with that of the conventional packed bed using the dispersion-withdead-region model. Although a slight increase in gas-side pressure drop was observed, the new packing arrangement showed some improvement in Peclet number and liquid holdup compared to the conventional packed bed. The improvement was significant at low flow rates and was due to better liquid distribution induced by the intermittent voids in the column.

Introduction

The efficiency of many significant industrial processes is dependent upon good contact between the gas and liquid phases. Among the major items of equipment available for this purpose are packed columns, commonly employed in a diversity of engineering applications. One serious disadvantage of packed columns is the channelling of one phase through the other. This often gives regions of dry packing, inadequate distribution of liquid, and thus, ineffective mass transfer. It has been generally known that this occurs due to the high voidage near the wall as shown by Roblee et al. (1958), Ridgeway and Tarbuck (19671, and 0019-7874/79/1018-0256$01.00/0

Pillai (1977). The high voidage near the wall in the gas-liquid countercurrent packed columns leads to down-flow of liquid near the wall and up-flow of gas through the center of the bed. This effect is known as gross channelling and was verified experimentally by Dutkai and Ruckenstein (1968). The flow maldistribution in packed columns has been studied systematically in recent years. Stanek and Szekely (1972, 1974) developed a formulation for the flow of an incompressible fluid through two- and three-dimensional packed columns having regions of differing porosities. Their results indicate that a radial variation in resistance 0 1979 American Chemical Society

Catalyst Deactivation by Active Site Coverage and Pore Blockage

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